On M-structure, the asymptotic-norming property and locally uniformly rotund renormings

Abstract

Letr, s ∈ [0, 1], and let X be a Banach space satisfying the M(r, s)-inequality, that is, $\parallel x^{***} \parallel \geqslant r\parallel \pi _X x^{***} \parallel + s\parallel x^{***} - \pi _X x^{***} \parallel for x^{***} \in X^{***} ,$ where π_X is the canonical projection from X^*** onto X^*. We show some examples of Banach spaces not containing c₀, having the point of continuity property and satisfying the above inequality for r not necessarily equal to one. On the other hand, we prove that a Banach space X satisfying the above inequality for s=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition, X satisfies $\mathop {\lim \sup }\limits_\alpha \parallel u^* + sx_\alpha ^* \parallel \leqslant \mathop {\lim \sup }\limits_\alpha \parallel v^* + x_\alpha ^* \parallel $ whenever u^*, v^* ∈X^* with ‖u^*‖≤‖v^*‖ and (x ${}_{α}^{*}$ ) is a bounded weak^* null net in X^*, then X can be renormed to satisfy the, M(r, 1) and the M(1, s)-inequality such that X^* has the weak^* asymptotic-norming property I with respect to B_X.